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So, again, the root of 2,116 is thus found :
The greatest root in 21 is 4.
Let n be the other. 516 And as before (80 + n) x n = 516. I say then, how many times will 8 go in 51 ? And the answer is 6. And I say 86 x 6
516, the number sought, and the root is 46. In these cases the process is easy, for the squares are perfect squares, of only four places of figures.
Let us try 998,001. I mark it off as before, 998,00i, and supposing for the moment, that it consists of only four places of figures, I proceed as before :
::. proceeding as before, I say,
Now 180 will go in 1,880 1,701
10 times, but as the n must
be a digit, I try 9; and 189
с в н Now, looking back at Fig. 2, we may suppose that the 99 so found is A B, and we may suppose that A B is extended to
Fig. 4. H, and that the figure is constructed thus :-(Fig. 4.)
But we should really proceed as we had done before, for A E is now what A D was before, and B I F is what CEG was before.
And I say (1,980 + n) * n = 17,901, and as 19 will go in 179 9 times, I say, 1,989 x 9 17,901. Here, as the square was a perfect square, the root is also perfect, 999. And here it may be observed, that if the root had consisted of
any number of figures, the process must have been continued as long as it was necessary; for when the value of the second square is found, it is only to consider it as the square already found, in the first case, and to proceed with a continual repetition of what had been before done. When A E is found, we treat it as we did A D, and A I (Fig. 4) will, in its turn, be dealt with in the same manner. In these cases, the root has been actually found, because the squares were perfect, but where this is not the case, the decimal value of the root may be found by the same process, and be continued to as many places of decimals as the case may require.
There was a square grass plat, of which the side was 100 feet; what was the diagonal ? or how much would
100 a person save by walking across it? By a known property of right angled triangles, the square of CA is equal to the sum of the squares of the two sides.
A C2 = 1 00 00
1 00 00
(20 + n) х п = 100
60,400 56,564 28,282 x 2
56,564 3,836 Here we shall never get at finite answer, but the process may be carried on to as many places of decimals as may be deemed necessary.
58,57, &c. In looking back at Fig. 3, it is probable that the steps expressed algebraically, are more clear than any other to the eye of the scholar. The process of multiplying a + b by a + b, is as follows:
(a + b) x a = a + a b
a b + 62
aa + 2 a b + b2 If the scholar does not see this, he will do well to try several examples in numbers, e.g. 8 x 8 = 64 (5 + 3) x 5 25 + 15 (5 + 3) x 3
15 + 9
25 + 30 + 9
EXPLANATION OF THE CUBE ROOT.
If we wish to discover the solid contents of any regular figure, as of a room, we must multiply the length by the breadth, and that by the height. But in a cube, these quantities are all equal to each other. The cube of 1 is
1 1 x 1 x 1 2
8 2 x 2 x 2 3
27 3 x 3 x 3 4
64 4 x 4 x 4 5
125 = 5 x 5 x And so on
729 9 x 9 x 9 10
1,000 10 x 10 x 10 99
979,299 99 x 99 x 100
1,000,000 = 100 x 100 x 100 Here we see that each figure in the root never gives more than three places of figures in the cube ; therefore, where the object is to extract the root, we mark off every third figure, and not every second, as in the case of the square root.
If we consider the nature of a cube, and refer to Fig. 3, we shall observe that the base of the cube will be a square, composed of four parts, and that the thickness will be composed also of two parts, so that the whole will consist of eight parts. It will be (30 x 30) + 2 + (30 x 5) + (5 * 5) multiplied by (30 + 5) or (30 * 30 * 30) + 2 * (30 x 30 x 5) + (30 x 5 x5.)
(30 x 30 x 5.) + 2 * (30 x 5 x 5) + (5 * 5 * 5.)
(a + b)2
a + 3 aʻb + 3 ab+ 63.
15,875 In this case a is 30; therefore 3 a b 2,700 * 6.
The question, then, which I ask myself is, how many times will 27 go in 158?
The answer is, 5 times. The 5 must be the greatest possible number, but it may be too great-but I try it.
3 x a“ x 6 13,500 3 x 30" X 5
2,250 3 x 30 x 52
Let us take a large number which is not an exact cube
59456 In this case a is 40; therefore 3 x a b 4,800 x 6.
The question, then, which I ask myself is, how many times will 48 go in 594 ?
The answer is, 12 times. From the nature of the case, I know that b cannot be greater than 9, or a would have been 5 and not 4; therefore I try 9.
3 a^ b. 43,200
3 a 62
5,807,789 Now, a is 490; therefore 3 ao b 720,300 x b.
The question, then, which I ask myself is, how many times will 72 go in 580 ? The answer is 8,* but I try 7, for it will only just go 8 times.
3 a 6 5,042,100
693,316 The answer in whole numbers therefore is 497, and the cube of 497 is 122,763,473, which differs from the original number by the present remainder.
If the answer thus found be not sufficiently accurate for the practical question which we wish to answer, we must add 'three ciphers to the remainder, and proceed, as before, to find the first place in the decimals.
It is obvious that this whole proceeding is a very laborious as all such questions can be much more easily answered by logarithms, the extraction of the cube and square roots can only be useful as an exercise for arithmeticians. A practical man, who needed to use them
* With 8, the next subtraction would have been 5,856,992, which would have been too great.
frequently, and who had no knowledge of logarithms, would construct or obtain a table of roots sufficiently extensive for his own use.
A master who only required questions might easily make them for himself, by actually raising the powers; but he would find it more beneficial to his scholars, if he carried them on to higher branches of knowledge, rather than employed them in working sums, which contain more of difficulty than instruction.
Ertracts from Charges.
(Continued from page 18.)
UNANIMITY OF CHURCHMEN WITH REGARD TO EDUCATION.
That our confidence in the principle here asserted upon this important question, is neither misplaced nor presumptuous, we have much stronger grounds for believing, than we could possess if it were the result of any individual judgment. But upon what other point, within the recollection of any now living, has the church of England exhibited such perfect unanimity? From the metropolitans of our provinces, descending through all the various orders of the clergy, I have not heard that there has been a second opinion expressed : and they have carried their flocks with them in an agreement quite extraordinary upon this point; that the only system of education either safe or becoming for the members of the church of England to be placed under, is a system which admits them at all times unrestrictedly to receive, under the direction of their clergy, instruction in the holy Scriptures, and in the creeds, catechisms, and other authorised formularies of the Christian faith, as it was preached to the world by the apostles, and through God's great mercy, restored to us at the reformation. Neither is the unanimity here spoken of confined to England. It has been expressed with equal earnestness of feeling by the clergy and laity in every colonial diocese; so that we have the consent of our church throughout the world, recorded by an overwhelming majority against any such general scheme of education, as it seemed but recently in contemplation to impose upon it. Indeed, even more than this may be urged; for we have of late perceived that some dissenting bodies, convinced of the injustice and impracticability of such a scheme, have expressed themselves adverse to the attempt to introduce it.From the Lord Bishop of Australia's Charge, 1844.
THE STATE À GREAT GAINER BY CHURCH EDUCATION.
On the other hand, there are not wanting causes of regret and apprehension. I think (and wherefore should my opinion be suppressed ?) that there is very great reason to regret the extreme difficulties which have been thrown by our government in the way of obtaining aid for the support of schools, in cases where they were plainly wanted, and might by a very small outlay have been instituted and supported. It is little known to the public in general, what labours and privations have been undergone by many of the clergy in their endeavours to maintain schools through their own exertions, when public support was either withheld, or dispensed with much unwillingness in concession only to a naked claim of right, such as the very extreme letter of the regulations could not be so interpreted as to resist. There is, indeed, a mode of dealing out the public bounty amounting to a discouragement, which the meek in spirit, and they who set a becoming estimate on their own position and expectations, are equally incapable of encountering; and this, I know from experience, is the feeling which the government regulations for granting aid to schools, have,